Cube Tumbleweeding on a Plane Am not a mathematician so please forgive any ambiguity in construction of the question below.
Given:


*

*a cube, with faces $a_1$,... $a_6$, with $a_1$ and $a_6$ forming opposite sides of the cube, and face $a_1$ resting on the $x$-$y$ plane, 

*and with a transformation $r(p, q)$ defined as a $90$ degree rotation around a side between adjacent faces $a_p$ and $a_q$ such that at least one of the cube's faces is always touching the $x$-$y$ plane:


Does there exist a series of transformations, $r(p_1, q_1)$, $r(p_2, q_2)$, ..., $r(p_n, q_n)$ such that:


*

*every face $(a_1, ..., a_6)$ shall touch the $x$-$y$ plane an equal number of times $(k)$ 

*at the end of which the center of the cube returns to its original location?


If so, what is the minimum possible value of $k$?
 A: Suppose you roll the cube 6 times such that each face is turned to the bottom once. Imagine also, that it leaves a trail of squares behind. Those squares could then be rolled up into a cube, so they form a net of the cube. There is however no cube net that is a loop, so it is not possible to return to the starting location already.
The next smallest number of rolls that could work is 12, where every face is visited twice. For simplicity we might as well visit all faces once with the first 6 rolls, and then roll back the cube in reverse to revisit the same squares and visit all 6 faces again.
On way to roll is in the compass directions NENENE and then back WSWSWS.
If the faces of the cube are denoted with D=Down (=$a_1$), U=Up (=$a_6$), L=Left, R=Right, F=Front, B=Back, where F is initially on the south side, then the cube starts on D and rolls onto the faces BRUFLD. When tracing back the moves, it visits them in reverse, rolling onto LFURB and then D again.
You can do the reverse trip as SWSWSW instead. Then the faces visited in the second half would be FLUBR and D.
In both cases the cube returns in the same orientation.
